WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 530 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 9 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(dbl(0)) -> mark(0) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(dbls(nil)) -> mark(nil) active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y))) active(sel(0, cons(X, Y))) -> mark(X) active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) active(indx(nil, X)) -> mark(nil) active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z))) active(from(X)) -> mark(cons(X, from(s(X)))) active(dbl1(0)) -> mark(01) active(dbl1(s(X))) -> mark(s1(s1(dbl1(X)))) active(sel1(0, cons(X, Y))) -> mark(X) active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z)) active(quote(0)) -> mark(01) active(quote(s(X))) -> mark(s1(quote(X))) active(quote(dbl(X))) -> mark(dbl1(X)) active(quote(sel(X, Y))) -> mark(sel1(X, Y)) active(dbl(X)) -> dbl(active(X)) active(dbls(X)) -> dbls(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(indx(X1, X2)) -> indx(active(X1), X2) active(dbl1(X)) -> dbl1(active(X)) active(s1(X)) -> s1(active(X)) active(sel1(X1, X2)) -> sel1(active(X1), X2) active(sel1(X1, X2)) -> sel1(X1, active(X2)) active(quote(X)) -> quote(active(X)) dbl(mark(X)) -> mark(dbl(X)) dbls(mark(X)) -> mark(dbls(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) indx(mark(X1), X2) -> mark(indx(X1, X2)) dbl1(mark(X)) -> mark(dbl1(X)) s1(mark(X)) -> mark(s1(X)) sel1(mark(X1), X2) -> mark(sel1(X1, X2)) sel1(X1, mark(X2)) -> mark(sel1(X1, X2)) quote(mark(X)) -> mark(quote(X)) proper(dbl(X)) -> dbl(proper(X)) proper(0) -> ok(0) proper(s(X)) -> s(proper(X)) proper(dbls(X)) -> dbls(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(indx(X1, X2)) -> indx(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(dbl1(X)) -> dbl1(proper(X)) proper(01) -> ok(01) proper(s1(X)) -> s1(proper(X)) proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2)) proper(quote(X)) -> quote(proper(X)) dbl(ok(X)) -> ok(dbl(X)) s(ok(X)) -> ok(s(X)) dbls(ok(X)) -> ok(dbls(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) indx(ok(X1), ok(X2)) -> ok(indx(X1, X2)) from(ok(X)) -> ok(from(X)) dbl1(ok(X)) -> ok(dbl1(X)) s1(ok(X)) -> ok(s1(X)) sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2)) quote(ok(X)) -> ok(quote(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(0) -> 0 encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(01) -> 01 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_dbl(x_1)) -> dbl(encArg(x_1)) encArg(cons_dbls(x_1)) -> dbls(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(cons_dbl1(x_1)) -> dbl1(encArg(x_1)) encArg(cons_s1(x_1)) -> s1(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_0 -> 0 encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_dbl1(x_1) -> dbl1(encArg(x_1)) encode_01 -> 01 encode_s1(x_1) -> s1(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(dbl(0)) -> mark(0) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(dbls(nil)) -> mark(nil) active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y))) active(sel(0, cons(X, Y))) -> mark(X) active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) active(indx(nil, X)) -> mark(nil) active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z))) active(from(X)) -> mark(cons(X, from(s(X)))) active(dbl1(0)) -> mark(01) active(dbl1(s(X))) -> mark(s1(s1(dbl1(X)))) active(sel1(0, cons(X, Y))) -> mark(X) active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z)) active(quote(0)) -> mark(01) active(quote(s(X))) -> mark(s1(quote(X))) active(quote(dbl(X))) -> mark(dbl1(X)) active(quote(sel(X, Y))) -> mark(sel1(X, Y)) active(dbl(X)) -> dbl(active(X)) active(dbls(X)) -> dbls(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(indx(X1, X2)) -> indx(active(X1), X2) active(dbl1(X)) -> dbl1(active(X)) active(s1(X)) -> s1(active(X)) active(sel1(X1, X2)) -> sel1(active(X1), X2) active(sel1(X1, X2)) -> sel1(X1, active(X2)) active(quote(X)) -> quote(active(X)) dbl(mark(X)) -> mark(dbl(X)) dbls(mark(X)) -> mark(dbls(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) indx(mark(X1), X2) -> mark(indx(X1, X2)) dbl1(mark(X)) -> mark(dbl1(X)) s1(mark(X)) -> mark(s1(X)) sel1(mark(X1), X2) -> mark(sel1(X1, X2)) sel1(X1, mark(X2)) -> mark(sel1(X1, X2)) quote(mark(X)) -> mark(quote(X)) proper(dbl(X)) -> dbl(proper(X)) proper(0) -> ok(0) proper(s(X)) -> s(proper(X)) proper(dbls(X)) -> dbls(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(indx(X1, X2)) -> indx(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(dbl1(X)) -> dbl1(proper(X)) proper(01) -> ok(01) proper(s1(X)) -> s1(proper(X)) proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2)) proper(quote(X)) -> quote(proper(X)) dbl(ok(X)) -> ok(dbl(X)) s(ok(X)) -> ok(s(X)) dbls(ok(X)) -> ok(dbls(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) indx(ok(X1), ok(X2)) -> ok(indx(X1, X2)) from(ok(X)) -> ok(from(X)) dbl1(ok(X)) -> ok(dbl1(X)) s1(ok(X)) -> ok(s1(X)) sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2)) quote(ok(X)) -> ok(quote(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(01) -> 01 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_dbl(x_1)) -> dbl(encArg(x_1)) encArg(cons_dbls(x_1)) -> dbls(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(cons_dbl1(x_1)) -> dbl1(encArg(x_1)) encArg(cons_s1(x_1)) -> s1(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_0 -> 0 encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_dbl1(x_1) -> dbl1(encArg(x_1)) encode_01 -> 01 encode_s1(x_1) -> s1(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(dbl(0)) -> mark(0) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(dbls(nil)) -> mark(nil) active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y))) active(sel(0, cons(X, Y))) -> mark(X) active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) active(indx(nil, X)) -> mark(nil) active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z))) active(from(X)) -> mark(cons(X, from(s(X)))) active(dbl1(0)) -> mark(01) active(dbl1(s(X))) -> mark(s1(s1(dbl1(X)))) active(sel1(0, cons(X, Y))) -> mark(X) active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z)) active(quote(0)) -> mark(01) active(quote(s(X))) -> mark(s1(quote(X))) active(quote(dbl(X))) -> mark(dbl1(X)) active(quote(sel(X, Y))) -> mark(sel1(X, Y)) active(dbl(X)) -> dbl(active(X)) active(dbls(X)) -> dbls(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(indx(X1, X2)) -> indx(active(X1), X2) active(dbl1(X)) -> dbl1(active(X)) active(s1(X)) -> s1(active(X)) active(sel1(X1, X2)) -> sel1(active(X1), X2) active(sel1(X1, X2)) -> sel1(X1, active(X2)) active(quote(X)) -> quote(active(X)) dbl(mark(X)) -> mark(dbl(X)) dbls(mark(X)) -> mark(dbls(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) indx(mark(X1), X2) -> mark(indx(X1, X2)) dbl1(mark(X)) -> mark(dbl1(X)) s1(mark(X)) -> mark(s1(X)) sel1(mark(X1), X2) -> mark(sel1(X1, X2)) sel1(X1, mark(X2)) -> mark(sel1(X1, X2)) quote(mark(X)) -> mark(quote(X)) proper(dbl(X)) -> dbl(proper(X)) proper(0) -> ok(0) proper(s(X)) -> s(proper(X)) proper(dbls(X)) -> dbls(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(indx(X1, X2)) -> indx(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(dbl1(X)) -> dbl1(proper(X)) proper(01) -> ok(01) proper(s1(X)) -> s1(proper(X)) proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2)) proper(quote(X)) -> quote(proper(X)) dbl(ok(X)) -> ok(dbl(X)) s(ok(X)) -> ok(s(X)) dbls(ok(X)) -> ok(dbls(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) indx(ok(X1), ok(X2)) -> ok(indx(X1, X2)) from(ok(X)) -> ok(from(X)) dbl1(ok(X)) -> ok(dbl1(X)) s1(ok(X)) -> ok(s1(X)) sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2)) quote(ok(X)) -> ok(quote(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(01) -> 01 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_dbl(x_1)) -> dbl(encArg(x_1)) encArg(cons_dbls(x_1)) -> dbls(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(cons_dbl1(x_1)) -> dbl1(encArg(x_1)) encArg(cons_s1(x_1)) -> s1(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_0 -> 0 encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_dbl1(x_1) -> dbl1(encArg(x_1)) encode_01 -> 01 encode_s1(x_1) -> s1(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(dbl(0)) -> mark(0) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(dbls(nil)) -> mark(nil) active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y))) active(sel(0, cons(X, Y))) -> mark(X) active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) active(indx(nil, X)) -> mark(nil) active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z))) active(from(X)) -> mark(cons(X, from(s(X)))) active(dbl1(0)) -> mark(01) active(dbl1(s(X))) -> mark(s1(s1(dbl1(X)))) active(sel1(0, cons(X, Y))) -> mark(X) active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z)) active(quote(0)) -> mark(01) active(quote(s(X))) -> mark(s1(quote(X))) active(quote(dbl(X))) -> mark(dbl1(X)) active(quote(sel(X, Y))) -> mark(sel1(X, Y)) active(dbl(X)) -> dbl(active(X)) active(dbls(X)) -> dbls(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(indx(X1, X2)) -> indx(active(X1), X2) active(dbl1(X)) -> dbl1(active(X)) active(s1(X)) -> s1(active(X)) active(sel1(X1, X2)) -> sel1(active(X1), X2) active(sel1(X1, X2)) -> sel1(X1, active(X2)) active(quote(X)) -> quote(active(X)) dbl(mark(X)) -> mark(dbl(X)) dbls(mark(X)) -> mark(dbls(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) indx(mark(X1), X2) -> mark(indx(X1, X2)) dbl1(mark(X)) -> mark(dbl1(X)) s1(mark(X)) -> mark(s1(X)) sel1(mark(X1), X2) -> mark(sel1(X1, X2)) sel1(X1, mark(X2)) -> mark(sel1(X1, X2)) quote(mark(X)) -> mark(quote(X)) proper(dbl(X)) -> dbl(proper(X)) proper(0) -> ok(0) proper(s(X)) -> s(proper(X)) proper(dbls(X)) -> dbls(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(indx(X1, X2)) -> indx(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(dbl1(X)) -> dbl1(proper(X)) proper(01) -> ok(01) proper(s1(X)) -> s1(proper(X)) proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2)) proper(quote(X)) -> quote(proper(X)) dbl(ok(X)) -> ok(dbl(X)) s(ok(X)) -> ok(s(X)) dbls(ok(X)) -> ok(dbls(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) indx(ok(X1), ok(X2)) -> ok(indx(X1, X2)) from(ok(X)) -> ok(from(X)) dbl1(ok(X)) -> ok(dbl1(X)) s1(ok(X)) -> ok(s1(X)) sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2)) quote(ok(X)) -> ok(quote(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(01) -> 01 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_dbl(x_1)) -> dbl(encArg(x_1)) encArg(cons_dbls(x_1)) -> dbls(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(cons_dbl1(x_1)) -> dbl1(encArg(x_1)) encArg(cons_s1(x_1)) -> s1(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_0 -> 0 encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_dbl1(x_1) -> dbl1(encArg(x_1)) encode_01 -> 01 encode_s1(x_1) -> s1(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence dbls(mark(X)) ->^+ mark(dbls(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / mark(X)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(dbl(0)) -> mark(0) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(dbls(nil)) -> mark(nil) active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y))) active(sel(0, cons(X, Y))) -> mark(X) active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) active(indx(nil, X)) -> mark(nil) active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z))) active(from(X)) -> mark(cons(X, from(s(X)))) active(dbl1(0)) -> mark(01) active(dbl1(s(X))) -> mark(s1(s1(dbl1(X)))) active(sel1(0, cons(X, Y))) -> mark(X) active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z)) active(quote(0)) -> mark(01) active(quote(s(X))) -> mark(s1(quote(X))) active(quote(dbl(X))) -> mark(dbl1(X)) active(quote(sel(X, Y))) -> mark(sel1(X, Y)) active(dbl(X)) -> dbl(active(X)) active(dbls(X)) -> dbls(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(indx(X1, X2)) -> indx(active(X1), X2) active(dbl1(X)) -> dbl1(active(X)) active(s1(X)) -> s1(active(X)) active(sel1(X1, X2)) -> sel1(active(X1), X2) active(sel1(X1, X2)) -> sel1(X1, active(X2)) active(quote(X)) -> quote(active(X)) dbl(mark(X)) -> mark(dbl(X)) dbls(mark(X)) -> mark(dbls(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) indx(mark(X1), X2) -> mark(indx(X1, X2)) dbl1(mark(X)) -> mark(dbl1(X)) s1(mark(X)) -> mark(s1(X)) sel1(mark(X1), X2) -> mark(sel1(X1, X2)) sel1(X1, mark(X2)) -> mark(sel1(X1, X2)) quote(mark(X)) -> mark(quote(X)) proper(dbl(X)) -> dbl(proper(X)) proper(0) -> ok(0) proper(s(X)) -> s(proper(X)) proper(dbls(X)) -> dbls(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(indx(X1, X2)) -> indx(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(dbl1(X)) -> dbl1(proper(X)) proper(01) -> ok(01) proper(s1(X)) -> s1(proper(X)) proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2)) proper(quote(X)) -> quote(proper(X)) dbl(ok(X)) -> ok(dbl(X)) s(ok(X)) -> ok(s(X)) dbls(ok(X)) -> ok(dbls(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) indx(ok(X1), ok(X2)) -> ok(indx(X1, X2)) from(ok(X)) -> ok(from(X)) dbl1(ok(X)) -> ok(dbl1(X)) s1(ok(X)) -> ok(s1(X)) sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2)) quote(ok(X)) -> ok(quote(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(01) -> 01 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_dbl(x_1)) -> dbl(encArg(x_1)) encArg(cons_dbls(x_1)) -> dbls(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(cons_dbl1(x_1)) -> dbl1(encArg(x_1)) encArg(cons_s1(x_1)) -> s1(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_0 -> 0 encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_dbl1(x_1) -> dbl1(encArg(x_1)) encode_01 -> 01 encode_s1(x_1) -> s1(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(dbl(0)) -> mark(0) active(dbl(s(X))) -> mark(s(s(dbl(X)))) active(dbls(nil)) -> mark(nil) active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y))) active(sel(0, cons(X, Y))) -> mark(X) active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) active(indx(nil, X)) -> mark(nil) active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z))) active(from(X)) -> mark(cons(X, from(s(X)))) active(dbl1(0)) -> mark(01) active(dbl1(s(X))) -> mark(s1(s1(dbl1(X)))) active(sel1(0, cons(X, Y))) -> mark(X) active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z)) active(quote(0)) -> mark(01) active(quote(s(X))) -> mark(s1(quote(X))) active(quote(dbl(X))) -> mark(dbl1(X)) active(quote(sel(X, Y))) -> mark(sel1(X, Y)) active(dbl(X)) -> dbl(active(X)) active(dbls(X)) -> dbls(active(X)) active(sel(X1, X2)) -> sel(active(X1), X2) active(sel(X1, X2)) -> sel(X1, active(X2)) active(indx(X1, X2)) -> indx(active(X1), X2) active(dbl1(X)) -> dbl1(active(X)) active(s1(X)) -> s1(active(X)) active(sel1(X1, X2)) -> sel1(active(X1), X2) active(sel1(X1, X2)) -> sel1(X1, active(X2)) active(quote(X)) -> quote(active(X)) dbl(mark(X)) -> mark(dbl(X)) dbls(mark(X)) -> mark(dbls(X)) sel(mark(X1), X2) -> mark(sel(X1, X2)) sel(X1, mark(X2)) -> mark(sel(X1, X2)) indx(mark(X1), X2) -> mark(indx(X1, X2)) dbl1(mark(X)) -> mark(dbl1(X)) s1(mark(X)) -> mark(s1(X)) sel1(mark(X1), X2) -> mark(sel1(X1, X2)) sel1(X1, mark(X2)) -> mark(sel1(X1, X2)) quote(mark(X)) -> mark(quote(X)) proper(dbl(X)) -> dbl(proper(X)) proper(0) -> ok(0) proper(s(X)) -> s(proper(X)) proper(dbls(X)) -> dbls(proper(X)) proper(nil) -> ok(nil) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) proper(indx(X1, X2)) -> indx(proper(X1), proper(X2)) proper(from(X)) -> from(proper(X)) proper(dbl1(X)) -> dbl1(proper(X)) proper(01) -> ok(01) proper(s1(X)) -> s1(proper(X)) proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2)) proper(quote(X)) -> quote(proper(X)) dbl(ok(X)) -> ok(dbl(X)) s(ok(X)) -> ok(s(X)) dbls(ok(X)) -> ok(dbls(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) indx(ok(X1), ok(X2)) -> ok(indx(X1, X2)) from(ok(X)) -> ok(from(X)) dbl1(ok(X)) -> ok(dbl1(X)) s1(ok(X)) -> ok(s1(X)) sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2)) quote(ok(X)) -> ok(quote(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(nil) -> nil encArg(01) -> 01 encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_dbl(x_1)) -> dbl(encArg(x_1)) encArg(cons_dbls(x_1)) -> dbls(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(cons_dbl1(x_1)) -> dbl1(encArg(x_1)) encArg(cons_s1(x_1)) -> s1(encArg(x_1)) encArg(cons_sel1(x_1, x_2)) -> sel1(encArg(x_1), encArg(x_2)) encArg(cons_quote(x_1)) -> quote(encArg(x_1)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_0 -> 0 encode_mark(x_1) -> mark(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_from(x_1) -> from(encArg(x_1)) encode_dbl1(x_1) -> dbl1(encArg(x_1)) encode_01 -> 01 encode_s1(x_1) -> s1(encArg(x_1)) encode_sel1(x_1, x_2) -> sel1(encArg(x_1), encArg(x_2)) encode_quote(x_1) -> quote(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST